On terminal delta-wye reducibility of planar graphs

Lecture Notes in Computer Science, 2006

We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs.

Lecture Notes in Computer Science, 2007

We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most 11 log 2 n + 43. As a consequence, a canonic form of such graphs is computable in AC 1 by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in AC 1 . * Supported by an Alexander von Humboldt fellowship.

Journal of Combinatorial Theory, Series B, 1981

In the first paper 131, the author, together with Fiorini. has shown that maximal planar graphs are recognizable from their decks of vertex-deleted subgraphs. The aim of this paper is to show that such graphs are reconstructible. We first define an ordinary vertex to be a vertex whose valency is at least 4. Now, given a maximal planar graph G of minimum valency 3. we can recognize the maximal planarity of G from the deck g(G) 13). Therefore for any ordinary vertex v, we need only consider the p(v)-representations of G

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1989

Abstmct-In this paper we present two O ( n * ) planarization algorithms-PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on Lempel, Even, and Cederbaum's planarity testing algorithm [9] and its implementation using PQ-trees [8]. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. This algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G,, of a nonplanar graph G, algorithm MAXIMAL-PLANARIZE constructs a maximal planar subgraph of G which contains G,,. This latter algorithm can also be used to maximally planarize a biconnected planar graph.

2006

We show that computing the lexicographically first four-coloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P = NP. We discuss this application to non-self-reducibility and provide a general related result. We also discuss when raising a problem's NP-hardness lower bound to ∆ p 2hardness can be valuable.

We present an alternative linear time algorithm that computes an ordering that produces a fill-in that is minimal with respect to the subset relation. It is simpler than the algorithm in [6] and is easily parallelizable. The algorithm does not rely on the computation of a breadth-first search tree.

Discrete Mathematics, 2009

We show that if H is an odd-cycle, or any non-bipartite graph of girth 5 and maximum degree at most 3, then planar H-COL is NP-complete.

Discrete Applied Mathematics, 2000

A graph is called v-pancyclic if it contains a cycle of length l containing a given vertex v for 36l6n, and a graph G is called vertex pancyclic if G is v-pancyclic for all v. In this paper, we show that it is NP-complete to determine whether a 3-connected cubic planar graph is v-pancyclic for given vertex v, it is NP-complete to determine whether a 3-connected cubic planar graph is pancyclic, and it is NP-complete to determine whether a 3-connected planar graph is vertex pancyclic. We also show that every maximal outplanar graph is vertex pancyclic. ?

2000

A graph is path k-colorable if it has a vertex k-coloring in which the subgraph induced by each color class is a disjoint union of paths. A graph is path k-choosable if, whenever each vertex is assigned a list of k colors, such a coloring exists in which each vertex receives a color from its list.

The Electronic Journal of Combinatorics, 2018

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

Discrete Mathematics

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.

Journal of Graph Algorithms and Applications, 2013

A graph is B k -VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs constitute a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n 3/2 ) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we obtain a new proof that bipartite planar graphs are a subclass of 2-DIR.

Journal of Graph Theory, 2009

The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be N P-complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains N P-complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is N P-complete for planar graphs with girth five. The reduction is from planar graph 3-colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching-cuts are described. These classes include claw-free graphs, co-graphs, and graphs A preliminary version of this

European Journal of Combinatorics, 2008

The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a(G) ≤ 3 for any planar graph G. In this paper we prove that a(G) ≤ 2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3.

SIAM Journal on Computing, 2013

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.